Chapter 5: Probability and Random Variables

Consider the random experiment of tossing a coin once. There are two possible outcomes for this experiment, namely a head (H) or a tail (T).

a. Repeat the random experiment five times- that is toss a coin five times- and record the information required in the following table. (The third and fourth columns are for running totals and running proportions, respectively)

b. Based on your five tosses, what estimate would you give for the probability of a head when this coin is tossed once? Explain your answer.

c. Now toss the coin five more times and continue recording in the table so that you now have entries for tosses \(1-10\). Based on your \(10\) tosses, what estimate would you give for the probability of a head when this coin is tossed once? Explain your answer.

d. Now toss the coin \(10\) more times and continue recording in the table so that you now have entries for tosses \(1-20\). Based on your \(20\) tosses, what estimate you would give for the probability of a head when this coin is tossed once? Explain your reasoning.

e. In view of your results in parts (b)-(d), explain why the frequentist interpretation cannot be used as the definitions of probability.

Roulette. An American roulette wheel contains 38 numbers, of which 18 are red, 18 are black, and 2 are green. When the roulette wheel is spun, the ball is equally likely to land on any of the 38 numbers. For a bet on red, the house pays even odds ( i.e., 1 to 1 ). What should the odds actually be to make the bet fair?

Cyber Affair. As found in USA TODAY, results of a survey byInternational Communications Researchrevealed that roughly 75% of adult women believe that a romantic relationship over the Internet while in an exclusive relationship in the real world is cheating. What are the odds against randomly selecting an adult female Internet user who believes that having a " cyber affair " is cheating?

Belmont Stakes.The Belmont Stakes is the third leg, after the Kentucky Derby and Preakness Stakes, of the Triple Crown of thoroughbred horseracing. The morning-line betting odds of the two favorites, Orb and Revolutionary, for the 2013 Belmont Stakes were 7 to 2 (against) and 5 to I (against), respectively. Based on the morning-line betting odds, determine the probability that the winner of the race would be

(a) Orb. (b) Revolutionary.

Cursing Your Computer. A study was conducted by the firm Coleman & Associates, Inc. to determine who curses at their computer. The results, which appeared in USA TODAY, indicated that 46% of people age 18-34 years have cursed at their computer. What are the odds against a randomly selected 18- to 34-year-old having cursed at his or her computer?

If a member is selected at random from a finite population, probabilities are identical to .

Lightning Casualties. An issue ofTravel + Leisure Golf magazine reported several facts about lightning. Here are three of them.

• The odds of an individual being struck by lightning in a year in the United States are about 280,000 to I (against).
• The odds of an individual being struck by lightning in a year in Florida-the state with the most golf courses-are about 80,000 to I (against).
• About 5% of all lightning fatalities occur on golf courses.

Based on these data, answer the following questions.

(a) What is the probability of a person being struck by lightning in a year in the United States? Express your answer as a decimal rounded to eight places.

(b) What is the probability of a person being struck by lightning in a year in Florida? Express your answer as a decimal rounded to seven decimal places. (c) If a person dies from being hit by lightning, what are the odds that the fatality did not occur on a golf course?

What type of graphical displays that are useful for portraying events and relationships among ?

Constract a venn diagram representing the event.

Part. (a) (not E).

Part. (b) (A or B)

Constract a venn diagram representing the event.

Part (a) $\left(A\&B\right)$.

Part (b)$\left(A\&B\hspace{0.17em}\&C\right)$.